Affine Parikh automata
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 46 (2012) no. 4, pp. 511-545.

The Parikh finite word automaton (PA) was introduced and studied in 2003 by Klaedtke and Rueß. Natural variants of the PA arise from viewing a PA equivalently as an automaton that keeps a count of its transitions and semilinearly constrains their numbers. Here we adopt this view and define the affine PA, that extends the PA by having each transition induce an affine transformation on the PA registers, and the PA on letters, that restricts the PA by forcing any two transitions on the same letter to affect the registers equally. Then we report on the expressiveness, closure, and decidability properties of such PA variants. We note that deterministic PA are strictly weaker than deterministic reversal-bounded counter machines.

DOI : 10.1051/ita/2012013
Classification : 68Q45
Mots-clés : automata, semilinear sets, affine functions, counter machines 
@article{ITA_2012__46_4_511_0,
     author = {Cadilhac, Micha\"el and Finkel, Alain and McKenzie, Pierre},
     title = {Affine {Parikh} automata},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {511--545},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {4},
     year = {2012},
     doi = {10.1051/ita/2012013},
     mrnumber = {3107862},
     zbl = {1279.68136},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita/2012013/}
}
TY  - JOUR
AU  - Cadilhac, Michaël
AU  - Finkel, Alain
AU  - McKenzie, Pierre
TI  - Affine Parikh automata
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2012
SP  - 511
EP  - 545
VL  - 46
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita/2012013/
DO  - 10.1051/ita/2012013
LA  - en
ID  - ITA_2012__46_4_511_0
ER  - 
%0 Journal Article
%A Cadilhac, Michaël
%A Finkel, Alain
%A McKenzie, Pierre
%T Affine Parikh automata
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2012
%P 511-545
%V 46
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ita/2012013/
%R 10.1051/ita/2012013
%G en
%F ITA_2012__46_4_511_0
Cadilhac, Michaël; Finkel, Alain; McKenzie, Pierre. Affine Parikh automata. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 46 (2012) no. 4, pp. 511-545. doi : 10.1051/ita/2012013. http://www.numdam.org/articles/10.1051/ita/2012013/

[1] B.S. Baker and R.V. Book, Reversal-bounded multipushdown machines. J. Comput. Syst. Sci. 8 (1974) 315-332. | MR | Zbl

[2] M. Blattner and M. Latteux, Parikh-bounded languages, in ICALP. Lect. Notes Comput. Sci. 115 (1981) 316-323. | MR | Zbl

[3] R. Book, M. Nivat and M. Paterson, Reversal-bounded acceptors and intersections of linear languages. SIAM J. Comput. 3 (1974) 283. | MR | Zbl

[4] F. Brandenburg, Analogies of PAL and COPY, in Fundamentals of Computation Theory. Lect. Notes in Comput. Sci. 117 (1981) 61-70. | MR | Zbl

[5] E. Chiniforooshan, M. Daley, O.H. Ibarra, L. Kari and S. Seki, One-reversal counter machines and multihead automata : revisited, in Proc. of SOFSEM. ACM (2011) 166-177. | MR | Zbl

[6] H.B. Enderton, A Mathematical Introduction to Logic. Academic Press (1972). | MR | Zbl

[7] J. Ferrante and C. Rackoff, A decision procedure for the first order theory of real addition with order. SIAM J. Comput. 4 (1975) 69-76. | MR | Zbl

[8] P. Ganty, R. Majumdar and B. Monmege, Bounded underapproximations. Form. Methods Syst. Des. 40 (2012) 206-231. | Zbl

[9] S. Ginsburg and E.H. Spanier, Semigroups, Presburger formulas and languages. Pacific J. Math. 16 (1966) 285-296. | MR | Zbl

[10] S. Ginsburg and E. Spanier, Finite-turn pushdown automata. SIAM J. Control Optim. 4 (1966) 429. | MR | Zbl

[11] S.A. Greibach, A note on undecidable properties of formal languages. Math. Syst. Theor. 2 (1968) 1-6. | Zbl

[12] O.H. Ibarra, Reversal-bounded multicounter machines and their decision problems. J. ACM 25 (1978) 116-133. | MR | Zbl

[13] O.H. Ibarra and J. Su, A technique for proving decidability of containment and equivalence of linear constraint queries. J. Comput. Syst. Sci. 59 (1999) 1-28. | MR | Zbl

[14] O.H. Ibarra, J. Su, Z. Dang, T. Bultan and R.A. Kemmerer, Counter machines and verification problems. Theor. Comput. Sci. 289 (2002) 165-189. | MR | Zbl

[15] W. Karianto, Parikh automata with pushdown stack. Diploma thesis, RWTH Aachen (2004).

[16] F. Klaedtke and H. Rueß, Parikh automata and monadic second-order logics with linear cardinality constraints. Universität Freiburg, Tech. Rep. 177 (2002).

[17] F. Klaedtke and H. Rueß, Monadic second-order logics with cardinalities, in Proc. of ICALP. Lect. Notes Comput. Sci. 2719 (2003) 681-696. | MR | Zbl

[18] S.Y. Kuroda, Classes of languages and linear bounded automata. Inform. Control 7 (1964) 207-223. | MR | Zbl

[19] M. Latteux, Mots infinis et langages commutatifs. RAIRO Inform. Théor. Appl. 12 (1978) 185-192. | Numdam | MR | Zbl

[20] D.R. Mazur, Combinatorics : A Guided Tour. Mathematical Association of Mathematics (2010). | MR | Zbl

[21] P. Mckenzie, M. Thomas and H. Vollmer, Extensional uniformity for boolean circuits. SIAM J. Comput. 39 (2010) 3186-3206. | MR | Zbl

[22] H. Seidl, T. Schwentick and A. Muscholl, Numerical document queries, in Principles of Database Systems. ACM, San Diego, CA, USA (2003) 155-166.

[23] H. Straubing, Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser, Boston (1994). | MR | Zbl

[24] P. Tesson and D. Thérien, Logic meets algebra : the case of regular languages. Log. Meth. Comput. Sci. 3 (2007) 1-37. | MR | Zbl

[25] L.P.D. Van Den Dries, Tame Topology and O-minimal Structures. Cambridge Univ. Press (1998). | MR | Zbl

[26] P. Wolper and B. Boigelot, An automata-theoretic approach to Presburger arithmetic constraints, in Static Analysis (SAS'95). Lect. Notes Comput. Sci. 983 (1995) 21-32.

[27] S.D. Zilio and D. Lugiez, Xml schema, tree logic and sheaves automata, in Rewriting Techniques and Applications (2003) 246-263. | MR | Zbl

Cité par Sources :